Let $n:=M$. Since you say "the denominator follows a Chi-square distribution with $2M$ degrees of freedom", it appears that you assume the $x_i$'s to be iid standard complex Gaussian; anyway, if you need $a>1$, then you can do a simple rescaling. 

Thus, let us assume indeed that the $x_i$'s are iid standard complex Gaussian. It is then clear that the distribution of $z$ in $\mathbb C=\mathbb R^2$ is rotation invariant. So, it is enough to find the distribution of $|z|^2$, which is the same as that of 
\begin{equation}
	R:=\frac X{(X+Y)^2},
\end{equation}
where $X$ and $Y$ are independent random variables such that $X\sim\text{Gamma}(1,2)$ and $Y\sim\text{Gamma}(n-1,2)$. 

Considering now the transformation from $(X,Y)$ to $(R,S)$ with $S:=X$ (say) and using the corresponding formula -- see e.g. formula (41) in [Transformations][1],
we obtain the joint density of $(R,S)$, say $f_{R,S}$. Then the density of $R$ is given by the formula 
\begin{equation}
	f_R(r)=\int_0^\infty f_{R,S}(r,s)\,ds. 
\end{equation}
Using in this integral the substitution $v=\sqrt{rs}$, we find 
\begin{equation}
	f_R(r)=\frac1{(n-2)!2^n r^{n+1}}\,\int_0^1 v^n(1-v)^{n-2}\exp\Big\{-\frac v{2r}\Big\}\,dv \tag{1}
\end{equation}
for $r>0$, with $f_R(r)=0$ for $r\le0$. 

Using the binomial expansion for $(1-v)^{n-2}$ and then integrating by parts, we can express the latter integral as a finite sum (with the number of summands depending on $n$) of elementary functions. 
For instance, for $n=3$ and $r>0$ 
\begin{equation}
	f_R(r)=\exp\Big\{-\frac1{2r}\Big\}\left(\frac{1}{2 r^2}+96 r+\frac{6}{r}+36\right)-96 r+12; 
\end{equation}
here is the graph of this function: 

[![enter image description here][2]][2]

However, by the dominated convergence theorem, the integral in (1) converges to $c_n:=\int_0^1 v^n(1-v)^{n-2}\,dv=n!(n-2)!/(2n-1)!$ as $r\to\infty$, and so, $f_R(r)\sim \frac{c_n}{(n-2)!2^n r^{n+1}}$ as $r\to\infty$. So, the tail of the distribution of $|z|^2$ is power-like, which shows that "the resulting distribution [of $z$]" is **not** Gaussian.  


  [1]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwj0g5uK6MTYAhVM6IMKHVUUD7AQFggqMAA&url=http%3A%2F%2Fwww2.econ.iastate.edu%2Fclasses%2Fecon671%2Fhallam%2Fdocuments%2FTransformations.pdf&usg=AOvVaw3g5QBi4bRJn_rXqfhWenlh
  [2]: https://i.sstatic.net/e6pRT.png